Given Expression: We are given the expression y = cos^(4)(x)sin^(4)(x). This means we need to find the value of y in terms of cos(x) and sin(x) raised to the fourth power.Using Trigonometric Identity: To simplify the expression, we can use the identity sin^2(x) + cos^2(x) = 1, which is a fundamental trigonometric identity. However, since we have fourth powers, we need to square the identity to use it effectively in our expression. (sin^2(x) + cos^2(x))^2 = 1^2Expanding Identity: Expanding the left side of the equation, we get: sin^4(x) + 2sin^2(x)cos^2(x) + cos^4(x) = 1 This is the squared version of the Pythagorean identity.Analyzing Original Expression: Now, we notice that our original expression y = cos^(4)(x)sin^(4)(x) is a product of sin^4(x) and cos^4(x), which are both present in the expanded identity. However, there is no direct simplification that can be made using this identity, as our expression does not contain the term 2sin^2(x)cos^2(x).Final Simplification: Since we cannot simplify the expression further using trigonometric identities, we conclude that y = cos^(4)(x)sin^(4)(x) is already in its simplest form in terms of cos(x) and sin(x).

Precalculus

Operations with rational exponents

Full solution

y=\cos^{4}x\sin^{4}x

Given Expression: We are given the expression $y = \cos^{4}(x)\sin^{4}(x)$ . This means we need to find the value of $y$ in terms of $\cos(x)$ and $\sin(x)$ raised to the fourth power.
Using Trigonometric Identity: To simplify the expression, we can use the identity $\sin^2(x) + \cos^2(x) = 1$ , which is a fundamental trigonometric identity. However, since we have fourth powers, we need to square the identity to use it effectively in our expression. $(\sin^2(x) + \cos^2(x))^2 = 1^2$
Expanding Identity: Expanding the left side of the equation, we get: $\newline$ $\sin^4(x) + 2\sin^2(x)\cos^2(x) + \cos^4(x) = 1$ $\newline$ This is the squared version of the Pythagorean identity.
Analyzing Original Expression: Now, we notice that our original expression $y = \cos^{4}(x)\sin^{4}(x)$ is a product of $\sin^{4}(x)$ and $\cos^{4}(x)$ , which are both present in the expanded identity. However, there is no direct simplification that can be made using this identity, as our expression does not contain the term $2\sin^{2}(x)\cos^{2}(x)$ .
Final Simplification: Since we cannot simplify the expression further using trigonometric identities, we conclude that $y = \cos^{4}(x)\sin^{4}(x)$ is already in its simplest form in terms of $\cos(x)$ and $\sin(x)$ .